Properties

Label 2-2151-1.1-c1-0-89
Degree $2$
Conductor $2151$
Sign $-1$
Analytic cond. $17.1758$
Root an. cond. $4.14437$
Motivic weight $1$
Arithmetic yes
Rational no
Primitive yes
Self-dual yes
Analytic rank $1$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 2.02·2-s + 2.10·4-s − 3.48·5-s + 1.37·7-s + 0.213·8-s − 7.05·10-s − 0.191·11-s + 4.24·13-s + 2.77·14-s − 3.77·16-s − 4.07·17-s − 5.16·19-s − 7.32·20-s − 0.388·22-s − 0.978·23-s + 7.11·25-s + 8.59·26-s + 2.88·28-s − 10.0·29-s − 4.04·31-s − 8.08·32-s − 8.26·34-s − 4.76·35-s + 2.15·37-s − 10.4·38-s − 0.744·40-s − 0.0317·41-s + ⋯
L(s)  = 1  + 1.43·2-s + 1.05·4-s − 1.55·5-s + 0.517·7-s + 0.0755·8-s − 2.23·10-s − 0.0577·11-s + 1.17·13-s + 0.741·14-s − 0.944·16-s − 0.989·17-s − 1.18·19-s − 1.63·20-s − 0.0827·22-s − 0.204·23-s + 1.42·25-s + 1.68·26-s + 0.545·28-s − 1.85·29-s − 0.727·31-s − 1.42·32-s − 1.41·34-s − 0.806·35-s + 0.353·37-s − 1.69·38-s − 0.117·40-s − 0.00496·41-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2151 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2151 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(2151\)    =    \(3^{2} \cdot 239\)
Sign: $-1$
Analytic conductor: \(17.1758\)
Root analytic conductor: \(4.14437\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{2151} (1, \cdot )$
Primitive: yes
Self-dual: yes
Analytic rank: \(1\)
Selberg data: \((2,\ 2151,\ (\ :1/2),\ -1)\)

Particular Values

\(L(1)\) \(=\) \(0\)
\(L(\frac12)\) \(=\) \(0\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 \)
239 \( 1 + T \)
good2 \( 1 - 2.02T + 2T^{2} \)
5 \( 1 + 3.48T + 5T^{2} \)
7 \( 1 - 1.37T + 7T^{2} \)
11 \( 1 + 0.191T + 11T^{2} \)
13 \( 1 - 4.24T + 13T^{2} \)
17 \( 1 + 4.07T + 17T^{2} \)
19 \( 1 + 5.16T + 19T^{2} \)
23 \( 1 + 0.978T + 23T^{2} \)
29 \( 1 + 10.0T + 29T^{2} \)
31 \( 1 + 4.04T + 31T^{2} \)
37 \( 1 - 2.15T + 37T^{2} \)
41 \( 1 + 0.0317T + 41T^{2} \)
43 \( 1 - 2.77T + 43T^{2} \)
47 \( 1 + 0.680T + 47T^{2} \)
53 \( 1 + 4.91T + 53T^{2} \)
59 \( 1 + 3.94T + 59T^{2} \)
61 \( 1 - 3.63T + 61T^{2} \)
67 \( 1 - 5.37T + 67T^{2} \)
71 \( 1 - 8.95T + 71T^{2} \)
73 \( 1 + 7.76T + 73T^{2} \)
79 \( 1 + 12.9T + 79T^{2} \)
83 \( 1 + 2.22T + 83T^{2} \)
89 \( 1 - 9.94T + 89T^{2} \)
97 \( 1 + 4.84T + 97T^{2} \)
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   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−8.530870157944931102819483038222, −7.84217838752792714741724178034, −6.96365506708676985243235252067, −6.20412731309315540232461804618, −5.31823844500458410180651922202, −4.29411094829731234065507109263, −4.04762707759668056951119204519, −3.22183158209565315723152687716, −1.94638282209047600344721312281, 0, 1.94638282209047600344721312281, 3.22183158209565315723152687716, 4.04762707759668056951119204519, 4.29411094829731234065507109263, 5.31823844500458410180651922202, 6.20412731309315540232461804618, 6.96365506708676985243235252067, 7.84217838752792714741724178034, 8.530870157944931102819483038222

Graph of the $Z$-function along the critical line